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\cl{\bf              About the Costa Minimal Surface    }                                   

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\cl{                               H. Karcher                         }


\lf
This surface was responsible for the re-kindling of interest
 in minimal surfaces in 1982. It is a minimal {\bf embedding} of
the 3-punctured square torus. Its planar symmetry lines cut
this surface into four conformal squares and the two straight
lines through the saddle are the diagonals of these squares.
Because of the emphasis on the symmetries, our formulas are
taken from [K2.]

\lf
    The Gauss map of such a surface is determined by its
qualitative properties only up to a multiplicative factor cc
which we suggest for the morphing (as in the Chen-Gackstatter
case).  It closes the period (at cc0) with an intermediate value
argument.


\lf
After Costa's existence discovery, Hoffman-Meeks proved
embeddedness; they also found a deformation family through
rectangular tori, where the middle end deforms from a planar
one to a caten\-oid end. They generalized this family to any genus
by increasing the dihedral symmetry.

\lf
[K2]  H. Karcher, Construction of minimal surfaces, in ``Surveys in
      Geometry'', Univ. of Tokyo, 1989, and Lecture Notes No. 12,
      SFB 256, Bonn, 1989, pp. 1--96.


\lf
  For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

\lf
[KWH]  H. Karcher, F. Wei, and D. Hoffman,\break The genus one helicoid, and
         the minimal surfaces that led to its discovery, in ``Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais'
         Sixtieth Birthday", K. Uhlenbeck Editor, Publish or Perish Press, 1993

\lf
[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab,
           Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991


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